# Internal division

Coordinates of the point

*C*which divides the line segment joining the points*A*(*x*_{1},*y*_{1}) and*B*(*x*_{2},*y*_{2}) internally in the ratio*m*:*n*are given by*x*=

# External division

Coordinates of the point

*C*which divides the line segment joining the points*A*(*x*_{1},*y*_{1}) and*B*(*x*_{2},*y*_{2}) externally in the ratio*m*:*n*are given by*x*=

*Notes:*- The midpoint of (
*x*_{1},*y*_{1}) and (*x*_{2},*y*_{2}) is . - The centroid (
*G*) (concurrency point of medians of a triangle) of a triangle having vertices (*x*_{1},*y*_{1}), (*x*_{2},*y*_{2}), and (*x*_{3},*y*_{3}) is*G*using the fact that centroid is concurrency point of medians, and*G*divides median*AD*in ratio 2:1. - Orthocenter of a triangle is point of concurrency of altitudes of triangle.
- In a triangle (not equilateral) orthocenter (H) (concurrency point of altitudes of triangle), centroid (
*G*), and circumcenter (*O*) are collinear and in the ratio*HG*/*GO*= 2. - If circumcenter of a triangle (concurrency point of perpendicular bisector of sides of triangle) with vertices (
*x*_{1},*y*_{1}), (*x*_{2},*y*_{2}), and (*x*_{3},*y*_{3}) is origin (0, 0) then orthocenter is (*x*_{1}+*x*_{2}+*x*_{3},*y*_{1}+*y*_{2}+*y*_{3}). - Incenter of a triangle (concurrency point of internal angle bisector) with vertices
*A*(*x*_{1},*y*_{1}),*B*(*x*_{2},*y*_{2}), and*C*(*x*_{3},*y*_{3}) is*I*using thefact that internal angle bisector*AD*divides*BC*in the ratio*AB*/*AC*.

**Excenters**

*A*(

*x*

_{1},

*y*

_{1}),

*B*(

*x*

_{2},

*y*

_{2}), and

*C*(

*x*

_{3},

*y*

_{3}) be the vertices of the triangle

*ABC*, and let

*a*,

*b*, and

*c*be the lengths of the sides

*BC*,

*CA*, and

*AB*respectively.

The circle which touches the sides

*BC*and two sides

*AB*and

*AC*produced is called the

**escribed circle**opposite to the angle

*A*.

The bisectors of the external angle

*B*and

*C*meet at a point

*I*

_{1}which is the center of the escribed circle opposite to the angle

*A*.

The coordinates of

*I*

**are given by**

_{1}*I*

**and**

_{2}*I*

**(centers of escribed circles opposite to the angles**

_{3}*B*and

*C*respectively) are given by

- To prove that
*A*,*B*,*C*,*D*are vertices of

Parallelogram | Show that diagonals AC and BD bisect each other. |

Rhombus | Show that diagonals AC and BD bisect each other and adjacent sides AB and BC are equal. |

Square | Show that diagonals AC and BD equal and bisect each other and adjacent sides AB and BC are equal. |

Rectangle | Show that diagonals AC and BD equal and bisect each other. |